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RESEARCH Open Access
Hop-distance relationship analysis with
quasi-UDG model for node localization in
wireless sensor networks
Deyun Gao
1
, Ping Chen
2
, Chuan Heng Foh
3*
and Yanchao Niu
1
Abstract
In wireless sensor networks (WSNs), location information plays an imp ortant role in many fundamental services
which includes geographic routing, target tracking, location-based coverage, topology control, and others. One
promising approach in sensor network localization is the determination of location based on hop counts. A critical
priori of this approach that directly influences the accuracy of location estimation is the hop-distance relationship.
However, most of the related works on the hop-distance relationship assume the unit-disk graph (UDG) model that
is unrealistic in a practical scenario. In this paper, we formulate the hop-distance relationship for quasi-UDG model
in WSNs where sensor nodes are randomly and independently deployed in a circular region based on a Poisson
point process. Different from the UDG model, quasi-UDG model has the non-uniformity property for connectivity.
We derive an approximated recursive expression for the probability of the hop count with a given geographic
distance. The border effect and dependence problem are also taken into consideration. Furthermore, we give the
expressions describing the distribution of distance with known hop counts for inner nodes and those suffered
from the border effect where we discover the insignificance of the border effect. The analytical results are
validated by simulations showing the accuracy of the employed approximation. Besides, we demonstrate the
localization application of the formulated relationship and show the accuracy improvement in the WSN
localization.
1 Introduction
In recent years, wireless sensor networks (WSNs) which
generally consist of a large number of small, inexpensive

and energy efficient sensor nodes have become one of
the most important and basic technologies for informa-
tion access [1]. WSNs have been widely used in military,
environment monitoring, medicine care, and transporta-
tion control. Spatial information is crucial for sensor
data to be interpreted meaningfully in many domains
such as environmental monitoring, smart building fail-
ure detection, and military target tracking. The location
information of sensors also helps facilitate WSN opera-
tion such a s routing to a geographic field of interests,
measuring quality of coverage, and achieving traffic load
balance. In many monitoring applications, the sensor
nodes must be aware its location to explain ‘what hap-
pens and where’.
While specialized localization devices exist such as
GPS, given the large number of sensor nodes involved
in building a single WSN, it is cost ineffectiv e to e quip
every sensor node with such a sophisticated device.
Therefore, seeking for an alternative localization tech-
nology in WSNs has become one major r esearch in
WSNs [2]. Over the past few years, many localization
algorithms have been proposed to provide sensor locali-
zation [3]. These localization protocols can be divided
into two categories: range-based and range-free . The
former is defined by methods that use absolut e point-
to-point distance estimates (range) or angle estimates
for computing locations. The latter makes no assump-
tion about the availability or validity o f such informa-
tion. Recently, range-free localization methods have
attracted much attention because no extra sophisticated

device for distance measurement is needed for each sen-
sor node. Despite the challenge in obtaining virtual
* Correspondence:
3
School of Computer Engineering, Nanyang Technological University,
639798, Singapore
Full list of author information is available at the end of the article
Gao et al. EURASIP Journal on Wireless Communications and Networking 2011, 2011:99
/>© 2011 Gao et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons At tribution
License ( which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly cited .
coordinates purely based on radio connectivity informa-
tion [4,5], attempts have been made in developing a
practical solution to achieve localization. A few repre-
sentative protocols of this range-free scheme include
DV-Hop [6], APIT [7] , DRLS [8], MDS-MAP [9], and
LS-SOM [10]. Most of the range-free localization
schemes, such as DV-Hop, need to compute the average
distance per hop to estimate a node’s locat ion. In other
words, the performance of these localization schemes
relies on the accuracy of the employed hop-distance
relationship. Since the determination of an accurate
hop-di stanc e relations hip depends on various complex
factors such as node deployment, node density, and
wireless communication technology that cannot be
easily quantified, the deduction process is tedious and
unlikely to produce an exact close form relationship
using, say the geometric methods [11].
Due to lack of any predetermined infrastructure and
self-organized nature, in most cases, the sensor

nodes are randomly and independently deployed in
a bounded area. For simplicity, the vast majority of
studies based on the idealized unit-disk graph (UDG)
network model, where any two sensors can directly
communicate with each other if and only if their geo-
graphic distance is smaller than a predetermined radio
range. Examples of these research include geo-routing
protocols [12,13], localization algorithms [8,14], and
topology control techniques [15,16]. Similarly, most of
the works related to the hop-distance relationship have
been investigated assuming the UDG model [11,17-23].
The probability that two randomly selected stations
with a k nown distance can communicate in K or less
hops with omnidirectional antennas has been analyzed
by Chandler [17]. Bettestetter and Eberspacher, derived
the probability of the distance of two randomly chosen
nodes deployed in a rectangular region within one or
two hops [18]. However, when the hop counts are lar-
ger than two, only simulation results are available. The
distribution parameters are computed by the iterative
formula which extends from [19] with a linear forma-
tion. Ekici et al. [20] studied the probability o f the k-
hop distance in two dimensional network based on the
approximated Gaussian distribution. Dulman et al. [11]
derived the relationship between the number of hops
separating two nodes and the physical distance
between them in one- and two-dimensional topologies
considering the UDG model. In the study, the approxi-
mated approach based on a Markov Chain in two-
dimensional case is rather complicated to compute.

Zhao and Liang [21] collected the hop-distance joint
distribution from Monte Carlo simulations in a circular
region and proposed an attenuated Gaussian approxi-
mation for the conditional probability distribution
function (pdf) of the Euclidean distance given a known
hop count. Ta et al. [22] provided a recursive equation
for the two rando mly located sensor nodes that are k-
hopneighborsgivenaknowndistanceinhomogeneous
wireless sensor networks. Ma et al. [23] proposed a
method to compute the conditional probability that
a destination node has hop-count h with respect to a
source node given that the distance between the source
and the destinat ion is d.
Despite the current efforts, no fixed communication
range exists in actual network environment for the rea-
sons such as multi-path fading and antenna issues.
Therefore, a certain level of deviation occurs between
the intended operation and actual operation in wireless
sensor networks when the UDG model is assumed in a
protocol design. To deal with this problem, a practical
model called the quasi Unit-disk Graph (quasi-UDG)
model is proposed recently [24]. The quasi-U DG model
can be characterized by two parameters, the radio range
R and the quasi-UDG factor a.Foranytwonodesin
the quasi-UDG model, if their distance is longer than R,
no direct communication link exists between the two.
Otherwise, if their distance is between aR and R, a com-
munication link exists with a probability of p
l
,andp

l
=
1 when their distance is shorter than aR. Given this
newly proposed practical property of connectivity, it
warrants an investigation of the hop-distance relation-
ship with the quasi- UDG model for the range-free loca-
lization schemes to capture practical connectivity
characteristics.
In this paper , we focus on exploiting the connectivity
property of the quasi-UDG model and analyze the rela-
tionship between the hop counts separating two nodes
and their geographic distance with a specific node den-
sity in a WSN. We seek approximation technique to
provide a scalable solution for the two-dimensional case.
We further demonstrate the application of the devel-
oped hop-distance relationship to a range-free localiza-
tion scheme.
In our WSN setup, we consider that sensor nodes are
deployed into a circular region S
b
with the radius R
b
,
where the deployment position follows a Poisson point
process with a certain density l. We set
p
l
=
α
1−α

(
R
d
− 1
)
such that a longer distance between two nodes has a
lower probability to form a direct communication link.
With this setup, we formulate the probability that a pair
of nodes with a known distance resulting a particular
hop count. Additionally, we also develop the probability
that a pair of nodes with a known distance gives a parti-
cular hop count. Finally, in our analysis, we present a
quantitative evaluation for the border effect of geo-
graphic distance distribution with a given hop count.
The rest of this paper is organized as follows. In
Section 2, we present our analytical model deriving an
approximate recursive formula for the hop-distance
Gao et al. EURASIP Journal on Wireless Communications and Networking 2011, 2011:99
/>Page 2 of 11
relationship considering the quasi-UDG model. Section
3 extends our analytical model by taking the border
effect and dependence problem into consi deration. Sec-
tion 4 formulates the probability distribution of distance
withknownhopcounts.InSection5,wedemonstrate
the use of our developed hop-distance relationship by
applying the relationship to a least squares (LS) based
localization algorithm. Finally, we report results in
Section 6 and draw important conclusions in Section 7.
2 The probability of the hop count given a
known distance

In general, the hop-distance relationship is influenced by
the density of sensor nodes and their deployment strat-
egy, as well as the radio communication characteristics.
Considering the more practical quasi-UDG model, it is
recognized that the formulation for the hop-distance
relationship with the considerati on of quasi-UDG model
is tedious and unlikely to produce an exact close form.
We seek approximation using a recursive approac h
to derive an approximated hop-distance relationship.
In this section, we focus on analyzing the probability
thataparticularpairofsensornodesformsacertain
hop count with a known distance.
Suppose that N sensor nodes are deployed randomly
in circular region S
b
with a radius R
b
.Thenumberof
nodes in any region is a Poisson random variable with
an average node density of
λ =
N
S
b
=
N
(π R
2
b
)

.Assumethat
the communication range of a node is R, the communi-
cation model between any pair of nodes follows the
quasi-UDG model with a factor of a where 0 < a <1.
With the quasi-UDG model, the communication area
between two nodes with the distance d can be further
divided into three cases shown as follows.
• If d ≤ aR, then the two nodes can communicate
directly.
• If aR<d≤ R, then the two nodes can communi-
cate with a probability p
l
, which is set to (R/d - 1)a/
(1 - a). It means that a longer distance betwee n two
nodes has a lower probability to form a direct com-
munication link.
• If d>R, then the two nodes cannot communicate
directly.
The quasi-UDG model is illustrated with an example
shown in Figure 1. In the figure, we assume that there
are two nodes u and v, their distance is d
uv
,andtheir
communication probability is P.LetF
h
(d) be the prob-
ability that a particular pair of nodes with d distance
apart is h hops a way from each other. In the following,
we shall first derive F
h

(d) for the case of h =1and
then h ≥ 2.
2.1 The case of h =1
For the case of h = 1, owing to the quasi-UDG model,
F
1
(d) is obviously

1
(d)=
⎧
⎨
⎩
1
α
1−α

R
d
− 1

0
d ≤ αR
αR < d ≤
R
d > R
(1)
2.2 The case of h ≥ 2
We first note that two nodes, named O
1

and O
2
,have
no direct link but may communicate through h - 1 relay
nodes. This gives rise to two possibilities, where
• O
2
is not the m-hop neighbor of O
1
if m<h.
• Within the communication range of O
2
,thereisa
least one (h - 1)-hop neighbor of O
1
that has a direct
link with O
2
.
For m<h, the probability, P
N
,thatO
2
is not the m-
hop neighbor of O
1
can be obtained as
P
N
=1−

h−1

m
=1

m
(d)
.
(2)
We shall now consider the second possibility in the fol-
lowing. Considering two circles which one centered at O
1
having a radius of r and the other centered at O
2
having
aradiusofR. We denote the distance between the two
centers as d and refer the common region of the two cir-
cles as S. The quantity P
r
(S) is defined as the probability
that in the area S, there is no (h - 1)-hop neighbor of O
1
that can communicate with O
2
directly. A differential
increment of dr on r can obtain a differential incremental
region of dS. Assume that the probability F
h
(d)ofany
pair of nodes is independent and statistically identical, we

αR R
P = 1
P = p
l
P = 0
P = p
l
P = 0
d
uv
<αR
αR < d
uv
<R αR<d
uv
<Rd
uv
>R d
uv
>R
u
Figure 1 Quasi-UDG model.
Gao et al. EURASIP Journal on Wireless Communications and Networking 2011, 2011:99
/>Page 3 of 11
have P
r
(S + dS)=P
r
(S)P
r

(dS). In the following subsec-
tions, we calculate P
r
(dS) based on three conditions,
which are d>R,
1+α
2
R < d <
R
,and
α
R < d
1+α
2
R
.
R",1,0,2,0,0pc,0pc,0pc,0pc>2.2.1 O
1
falls outside the
communication range of O
2
where d >R
In Figure 2, we see that dS can be further divided into
many differential regions rdrdθ. Since dr and dθ are infi-
nitesimal, the probability that there e xists more than
one sensor node in the region rdrdθ can be ignored,
and the probability that a single sensor node located
within rdrdθ can be approximated as lrdrdθ.
We term the circular region centered in O
2

with the
radius aR as
C
(
O
2
)
, and the annulus region centered in
O
2
with the l arger radius R and the smaller one aR as
A(
O
2
)
. There are two cases needed to be t aken into
consideration, which are
• When dS falls into
A
(
O
2
)
as shown in Figure 2(a),
r satisfies d-R≤ r ≤ d-aR or d + aR ≤ r ≤ d-R.
With the definition of the quasi-UDG model, every
differential region rdrdθ of dS has a corresponding
probability p
l
to communicate with O

2
. Therefore, P
r
(dS) is given by (3) where
P
r
(dS)=1− 2
h−1
(r)λrdr
ϕ

0
α
1 − α

R
l
− 1

dθ
.
(3)
As illustrated in Figure 2(a), we can get the following
relationship
ϕ = arccos
r
2
+ d
2
− R

2
2
rd
(4)
l =

r
2
+ d
2
− 2rd cos θ
.
(5)
• When dS covers both
C(
O
2
)
and
A
(
O
2
)
, r will be
bounded by d-aR ≤ r<d+ aR.Thepartrdrdθ
that falls within
C(
O
2

)
is surely a one-hop neighbor
of O
2
.Whenthatpartfallswithin
A
(
O
2
)
,ithasa
corresponding probability p
l
that it has a direct link
with O
2
. Then P
r
(dS) can be determined by
P
r
(dS)=1− 2
h−1
(r)λrdr
⎡
⎢
⎣
ϕ
1
+

ϕ

ϕ1
α
1 − α

R
l
− 1

dθ
⎤
⎥
⎦
(6)
and
ϕ
1
= arccos
r
2
+ d
2
− (αR)
2
2
rd
.
(7)
2.2.2 O

1
falls within the communication range of O
1
and d
satisfies
1+α
2
R < d <
R
We use the foregoing strategy for this derivation. We
notice that there a re three cases needed to be treated
individually which are given as follows.
• If 0 <r<R-d, dS will be the annulus region and the
entire section of dS will fall within
A(
O
2
)
,whichgives
P
r
(dS)=1− 2
h−1
(r)λrdr
π

0
α
1 − α


R
l
− 1

d
θ
(8)
• If R-d ≤ r<d-aR or d+aR ≤ r <R+d, dS will not
be the annulus region but the entire section of dS
will still fall within
A
(
O
2
)
.ThenwecanobtainP
r
(dS) by (3).
• If d-aR ≤ r<d+aR, dS will cover both
C(
O
2
)
and
A
(
O
2
)
. In this case, we can determine P

r
(dS) by (6).
2.2.3 O
1
falls within the communication range of O
2
and d
satisfies
α
R < d
1+α
2
R
There are four cases needed to be considered when O
1
falls within the communication range of O
2
and d
ϕ
rdrdθ
dS
θ
d
r
R
αR
O
1
O
2

(a)
ϕ
rdrdθ
1
rdrdθ
2
dS
θ
1
θ
2
d
r
R
αR
O
1
O
2
(
b
)
Figure 2 Illustration of dS when d>Rfor the case that (a) dS
locates in
A
(
O
2
)
, and (b) dS locates in

C
(
O
2
)
and
A
(
O
2
)
.
Gao et al. EURASIP Journal on Wireless Communications and Networking 2011, 2011:99
/>Page 4 of 11
satisfying the condition
αR < d
1+α
2
R
, which are
• If 0 <r<d-aR, dS will be the annulus region and
the entire section of dS will fall within
C(
O
2
)
.Then
we can determine P
r
(dS) by (8).

• If d-aR ≤ r <R-d, dS will still be the annulus region
but it covers both
C(
O
2
)
and
A
(
O
2
)
. Therefore, we
have
P
r
(dS)=1− 2
h−1
(r)λrdr

ϕ
1
+
π

ϕ
1
α
1 − α


R
l
− 1

dθ

(9)
• If R-d ≤ r <d+aR, dS will not be w ill the annulus
region and it covers both
C(
O
2
)
and
A
(
O
2
)
.The
probability P
r
(dS) can be obtained by (6).
• If d+aR ≤ r <R+d, dS will fall within the region
A(
O
2
)
, and hence we can compute P
r

(dS) by (3).
2.3 Determination of F
h
(d) for h ≥ 2
Consider that P
r
(dS) only depends on r with a specific d,
we set P
r
(dS)=1-g(r). From P
r
(S + dS)=P
r
(S)P
r
(dS),
the expression of P
r
(S) can be obtained by the following
linear differential equation where
P
r
(S) = exp
⎛
⎝
−
d+R

d
−R

g(r)dr
⎞
⎠
.
(10)
Therefore, with (2) and (10), the probability F
h
(d)
with h ≥ 2 can be obtained as

h
(d)=P
N
× (1 − P
r
(S))
=

1 −
h−1

i=1

i
(d)


1 − exp (−2λ(d))

(11)

where knowing d, Ω(d) can be determined by one o f
the following expressions, which are
• For d>hRor d<aR :

(
d
)
=0
;
(12)
• For R<d≤ hR :
(d)=

d−αR
d−R

h−1
(r)r
ϕ

0
α
1 − α

R
l
− 1

dθdr
+


d+αR
d−αR

h−1
(r)r
⎛
⎝
ϕ
1
+
ϕ

ϕ
1
α
1 − α

R
l
− 1

dθ
⎞
⎠
d
r
+

d+R

d+αR

h−1
(r)r
ϕ

0
α
1 − α

R
l
− 1

dθdr
(13)
• For
1+α
2
R < d ≤
R
:

(d)=

R−d
0

h−1
(r)r

π

0
α
1 − α

R
l
− 1

dθ dr
+

d−αR
R−d

h−1
(r)r
ϕ

0
α
1 − α

R
l
− 1

dθdr
+


d+αR
d−αR

h−1
(r)r(ϕ
1
+
ϕ

ϕ
1
α
1 − α

R
l
− 1

dθ )d
r
+

d+R
d+αR

h−1
(r)r
ϕ


0
α
1 − α

R
l
− 1

dθdr
(14)
• For
α
R < d ≤
1+α
2
R
:

(d)=

d−αR
0

h−1
(r)r
π

0
α
1 − α


R
l
− 1

dθ dr
+

R−d
d−αR

h−1
(r)r(ϕ
1
+
π

ϕ
1
α
1 − α

R
l
− 1

dθ)dr
+

d+αR

d−αR

h−1
(r)r(ϕ
1
+
ϕ

ϕ
1
α
1 − α

R
l
− 1

dθ)d
r
+

d+R
d+αR

h−1
(r)r
ϕ

0
α

1 − α

R
l
− 1

dθdr.
(15)
3 The border effect and dependence problem
In the above analysis, we do not consider borders of a
WSN. However, in a realistic scenario, the deployment
area of WSNs is finite and hence borders exist. It is
known that the probability F
h
(d) deri ved assuming that
both involved nodes are not near the border of a WSN
may give a slightly different result when one or both of
them fall near the border. This is known as the border
effect. One common handling of the border effect is to
consider the toroidal distance metric in the simulation
experiment where a node closed to the border can com-
municate directly with some nodes at the opposite border
[25]. While this special setup eliminates the border effect,
it creates discrepancy between the study and practical
setups which may lead to a certain level of errors.
Clearly, nodes which are closer to the border cover
smaller regions than those at least d away from the bor-
der, and therefore intuit ively the quantity for Ω(d)
should be smaller with the consideration of the border
effect. Apparently, the border effect gives a different

level of impacts in the measure of F
h
(d) with a different
distance between an involved node and the border.
Gao et al. EURASIP Journal on Wireless Communications and Networking 2011, 2011:99
/>Page 5 of 11
However, it is tedious to derive all cases considering the
border effect. For simplicity, we take two key cases of
the border effect into consideration. Assuming the cen-
ter of deployment area is O,weconsidertwoannulus
near the border in the following.
• The first annulus, called
A
1
(
o
)
, is between the cir-
cles with radius of R
b
-R and R
b
-aR.
• The second annulus, called
A
2
(
o
)
,isbetweenthe

circles with radius of R
b
-R and R
b
-aR.
We set an average metric ζ(h) which varies from 0 to
1 for each hop to determine the decrement of Ω(d). For
thecircleareawiththeradiusR
b
- R,whichcanbe
called
C(
o
)
, we can set ζ(h) = 1 accordingly.
Another factor we have to consider is the dependence.
The hop-distance relationship derived as aforesaid relies
on an implicit independence assumption, that is the prob-
ability F
h
(d) of any pair of nodes is in dependent and sta-
tistically identical. However as pointed in [22], the events
that those nodes with the direct link to O
2
are h - 1 hops
away from O
1
are not mutually independent for cases
when h>2, an d the calcul ation of F
h-1

(r)shouldinclude
appropriate dependence conditions. For example, as
shown in Figure 3, nodes O
1
and O
2
are d distance apart
and h hops away from each other where h = 3. The prob-
ability that node M
1
is a 2-hop neighbor of node O
1
is the
probability that there is at least one node located in the
area S
1
offering packet relay between nodes O
1
and M
1
.
Here, the area S
1
is the intersect area between the circles
with the centers O
1
and M
1
. Similarly, the probability that
node M

2
is a 2-hop neighbor of node O
1
is the probability
that there is at least one node located in the area S
2
which
can directly communicate with nodes O
1
and M
2
.Here,
the area S
2
is the intersect area between the circles with
the centers O
1
and M
2
. It is obvious in the figure that the
areas S
1
and S
2
share a common area S
12
indicating that
the calculated probabilities are not independent.
To include the impact of the dependence, we add a
new factor, namely ξ(h), into the expression of Ω(d).

Both factors ζ(h)andξ(h)areaddedtoallowΩ( d)to
reflect a practical setup, and they can be estimated by
statistical results via experiments. With the inclusion of
ζ(h) and ξ(h) into the expression of ω(h), (11) becomes

h
(d)=

1 −
h−1

i=1

i
(d)


1 − exp ( −2λω(h)(d))

.
(16)
4 Distance distribution with known hop counts
In this section, assume that sensor nodes are randomly
deployed in a circular region, we derive equations to
determine the probability density function of distance d
with a known hop count
f
H
(
d

)
.
Theorem 4.1 The probability density function for the
distance d between two nodes randomly deployed in a
circular region with the radius R
b
is
f
D
(
d
)
, where
f
D
(d)=
d
πR
4
b

4R
2
b
arccos

d
2R
b


− d

4R
2
b
− d
2

.
(17)
We provide the proof of Theorem 4.1 in A ppendix A.
According to Theorem 4.1, we can obtain the probabil-
ity density function of distance between any two nodes
in the areas
C
(
o
)
,
A
1
(
o
)
,and
A
2
(
o
)

. Their probabilit y
density functions of distance are
f
D
c
(d
)
,
f
D
A
1
(d
)
,and
f
D
A
2
(d
)
, respectively. We also term them as
f
D∗
(
d
)
,in
general, where the symbol * is appropriately substituted
by either

A
1
,
A
2
or
C
. Their expressions are given in
(18), (19) and (20) in the following.
f
D
A
1
(d)=
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
2
d
R
2
b

0 < d ≤ αR
2d
πRR
2
b
(1−α)(2R
2
b
−αR−R)
((R
b
, R
b
− αR,d) − π (R
b
− R)
2
) αR < d ≤ R
2d
πRR
2
b
(1−α)(2R
2
b
−αR−R)

(R
b
, R

b
− αR,d) − (R
b
, R
b
− R, d)

R < d ≤ 2R
b
− R
2d
πRR
2
b
(1−α)(2R
2
b
−αR−R)
(R
b
, R
b
− αR,d)2R
b
− R < d ≤ 2R
b
− α
R
(18)
f

D
A
2
(d)=
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
d
παRR
2
b
(2R
b
−αR)

4R
2
b
arccos(
d
2R
b

) − d

4R
2
b
− d
2
− 2π (R
b
− αR)
2

0 < d ≤ αR
d
παRR
2
b
(2R
b
−αR)

4R
2
b
arccos(
d
2R
b
) − d


4R
2
b
− d
2
− 2(R
b
, R
b
− αR, d)

αR < d ≤ 2R
b
− αR
d
παRR
2
b
(2R
b
−αR)

4R
2
b
arccos(
d
2R
b
) − d


4R
2
b
− d
2

2R
b
− αR < d ≤ 2R
b
(19)
f
D
C
(d)=
4d
π(R
b
−R)
2
arccos
d
2(R
b
−R)
−
4d
2
π(R

b
−R)
4

4(R
b
− R)
2
− d
2
s. t.0< d ≤ 2 ·
(
R
b
− R
)
(20)
where Λ(R, r, d) is given by
(R, r, d)=R
2
arccos
R
2
+d
2
−r
2
2dR
+ r
2

arccos
r
2
+d
2
−R
2
2dr
−
1
2

((r + R)
2
− d
2
)(d
2
− (R − r)
2
)
.
By the Bayes’ formula, given
f
D∗
(
d
)
and F
h

(d), we can
obtain the expression
f
H∗
(
d
)
which is the probability
density function of the geographical distance d when
the hop count h is known to be H*. This expression is
determined by
f
H∗
(d)=

h
(d)f
D∗
(d)

hR
r
0

h
(x)f
D∗
(x)dx
(21)
where r

0
= 0 when h = 1, and r
0
= aR when h >1.
S
1
S
2
S
12
O
1
O
2
M
1
M
2
d
Figure 3 Illustration of multihop-dependence problem.
Gao et al. EURASIP Journal on Wireless Communications and Networking 2011, 2011:99
/>Page 6 of 11
5 Localization Applications
With the development o f the hop-distance relationship
for the quasi-UDG model, in this section, we show
the application of this new relationship to a particular
localization algorithm using LS based localization
algorithms [26], and we call this newly designed locali-
zation algorithm enhance weighted least squares
(EWLS).

In a particular localization scenario in WSNs, we
assume that there is a number of nodes whose locations
are known, and they shall be called anchor nodes. Other
nodes that have no knowledge of their locations are
call ed unknown nodes. Consider that an unknown node
j can obtain the location x
i
, hop h
ji
and average hop-dis-
tance c
i
of an anchor node i. The distance betwee n
nodes j and i can be calculated as d
ji
= c
i
h
ji
.Inourtest
scenario, we place an anchor node o in the center and
add several other anchor nodes in the map.
We design a simple mechanism to compute the range
of distance d
ji
.Eachanchornodei collects some infor-
mation to other anchor node k, computes and ranks the
average hop-distance c
i(k)
= d

ik
/h
ik
,suchasc
i(1)
≥ c
i(2)
≥
≥ c
i(n)
. We set the range of average hop-distance as
c
i
=

n−1
k=1
||x
i
− x
(k)


n−1
k=1
h
i
(
k
)

≤ c
i
≤

n
k=2
||x
i
− x
(k)


n
k=2
h
i(k)
=
¯
c
i
.
(22)
Following that, the range of distance d
ji
can be com-
puted as
d
(M)
j
i

=
¯
c
i
× h
j
i
and
d
(m)
j
i
= c
i
× h
j
i
.Withthe
range of distance d
ji
,thevariancev
h
of the pdf
f
H
(
d
)
,
we compute the weights, w

i
, of measured distance d
ji
as
w
i
=
1
v
h

d
(M)
ji
d
(m)
j
i
f
H
(x)dx
.
(23)
Finally, we set W =diag(w
1
, ,w
n
)andcomputethe
location
ˆ

x
of an unknown node using the following
results, where
ˆx =(A
T
n
WA
n
)
−1
A
T
n
Wb
n
(24)
and
A
n
=2
⎡
⎢
⎣
x
1
− (x
i
) y
1
− (y

i
)
.
.
.
.
.
.
x
n
− (x
i
) y
n
− (y
i
)
⎤
⎥
⎦
b
n
=
⎡
⎢
⎣
x
2
1
− (x

2
i
)+y
2
1
− (y
2
i
)+(d
2
i
) − d
2
1
.
.
.
x
2
n
− (x
2
i
)+y
2
n
− (y
2
i
)+(d

2
i
) − d
2
n
⎤
⎥
⎦
(t )=

n
i=1
tw
i

n
i
=1
w
i
.
6 Result discussions
In this section, we compare the analytical and statisti-
cal results through simulat ion experiments to illustrate
the performance of our proposed hop-distance model.
To illustrate the benefit of applying our model to
LS-based localization algorithms, we compared our
enhanced algorithm of EWLS to two classical LS-
based localization algorithms namely LS [26] and
PDM [27].

6.1 Impacts of boarder effects and dependence
We first illustrate the impacts of the boarder effect and
dependence problem. In the experiments , we gather sta-
tisticsofthehopcountswithcorrespondingdistance
information using Monte Carlo simulations. All the
simulation data are collected from several scenarios
where N sensor nodes are randomly deployed in a circu-
lar region of radius R
b
, and the transmission range is set
to R with the consideration of the quasi-UDG model.
The parameters are set to N =400,R
b
= 200, R =50,a
= 0.75, and the r esult comparisons are listed in Table 1.
Let o be the deployment center. The region where
nodes are deployed away from the border is denoted as
C(
o
)
,andweterm
A
1
(
o
)
and
A
2
(

o
)
as the annulus
regions in which the distances to o are within (R
b
-R,
R
b
-aR] and (R
b
-aR, R
b
], respectively.
In Table 1 we use cumulativeabsolutedifference
(CAD) to measure the sum of absolute differences
between the analytical results and statistical data. We
set
CAD =

d
|
h
(d) − Sim
h
|
,whereF
h
(d)andSim
h
are the probabilities of two nodes giving a hop count of

h with a known distance of d obtained from the analysis
and simulation, respectively. Moreover, we denote CAD*
as the CAD measurement between analytical results
without the border effect consideration and statistical
data. For
A
1
(
o
)
and
A
2
(
o
)
, we can see that the CAD* of
each hop is larger than that o f CAD because of the
impact of the border effect.
Table 1 Comparisons between analytical and simulation
results of F
h
(d)
Hops 2 3 4 5 6 7 8 9
C(
o
)
CAD 0.34 0.36 0.85 1.49 2.12 2.76 3.36 3.9
ω(h) 1.0 0.77 0.70 0.65 0.63 0.60 0.58 0.54
A

1
(
o
)
CAD 0.42 0.38 0.86 1.52 2.13 2.69 3.31 3.98
CAD* 0.66 0.59 0.88 1.59 2.21 2.79 3.45 4.03
ω(h) 0.95 0.77 0.70 0.65 0.62 0.61 0.59 0.57
A
2
(
o
)
CAD 0.35 0.49 1.17 1.76 2.33 2.89 3.40 4.05
CAD* 0.74 0.75 1.19 1.85 2.45 3.06 3.61 4.16
ω(h) 0.92 0.77 0.69 0.65 0.62 0.61 0.59 0.58
Gao et al. EURASIP Journal on Wireless Communications and Networking 2011, 2011:99
/>Page 7 of 11
6.2 The validation of distribution of distance by a known
hop count
We conduct simulation experiments with N = 400, R
b
=
200, R = 50, a = 0.75 and present
f
H∗
(
d
)
in Figures 4, 5
and 6 with the statist ical data and our analytical results.

In all three cases, we note that the numerical results of
f
H∗
(
d
)
given in (21) show excellent agreement with the
simulation results. This excellent agreement confirms the
accuracy of our model for the estimation of the distance
given a known hop count between two sensor nodes.
6.3 Localization accuracy comparisons
In the following, we conduct several simulation experi-
ments to illustrate the performance of our proposed
EWLS algorithm. In the simulation, N = 100 sensor nodes
are randomly deployed in the circle
S
b
with the radius
R
b
= 200. The number of anchor nodes is 16 and the com-
munication range of each sensor node is R = 80. The fac-
tor a of the quasi-UDG model is set to 0.76. In Figure 7
(a), even within the communication range R of node 1, the
nodes 30, 38, 53, and 63 cannot communicate directly
with node 1 due to the considered quasi-UDG model.
With the network topology illustrated in Figure 7(a), we
show the localizati on errors of EWLS, LS, and PDM in
Figure 7. Apparently, the accuracy of EWLS is higher than
that of the t wo classical al gorithms where the average

localization errors of EWLS, LS, and PDM are 0.26702R,
0.29728R, and 0.28462R, respec tively. This confirms that
when WSNs exhibit the quasi-UDG connectivity behavior,
our new hop-distance relationship that captures the beha-
vior offers an improved accuracy in localization.
In the following, we further compare the localizatio n
accuracy among EWLS, LS and PDM under various sce-
narios. In these simulation experiments, we set N = 400,
and sensor nodes are deployed uniformly in the circle
area with the radius R
b
= 200. The connectivity of
nodes follows the quasi-UDG model. The localization
error is calculated as
ξ =

j
 x
j
−ˆx
j


(N − n
)
.
Firstly, we focus on the impact of the number of
anchor nodes. The factor a of quasi-UDG model is set
to 0.76 and the communication range R of each sensor
node is set to 50. In Figure 8, we can see that the locali-

zation error ξ of all three algorithms decreases with the
increase of number of anchor nodes. Among them, our
proposed EWLS always offers the best performance.
Secondly, we investigate the impact of the parameter
a of quasi-UDG model. In this scenario, we set the
number of anchor nodes to 40 and the parameter a var-
ies from 0.72 to 1. The localization error comparison is
given in F igure 9. We observe that when the parameter
a increases, the number of neighbor nodes increases
Figure 4 The distribution
f
H
C
(d
)
when the hop count falls
between 1 and 8.
Figure 5 The distribution
f
H
A
1
(d
)
.
Figure 6 The distribution
f
H
A
2

(d
)
.
Gao et al. EURASIP Journal on Wireless Communications and Networking 2011, 2011:99
/>Page 8 of 11
and the number of hops between an unknown node and
an anchor n ode decreases. Thus, the localization error
decreases, and our proposed EWLS algor ithm remains
the best among all for all considered a values.
Last we study the impact of the communication range
R of each sensor node. We set the parameter a of
quasi-UDG model to 0.76 and set the number of anchor
nodes to 40. Similarly, we compare the localization
errors in Figure 10 with a range of R values. We observe
that because the number of neighbor nodes of a node
increases when its communication range increases, and
number of hops between an unknown node and an
anchor decreases which leads to a decrease in localiza-
tion errors. Comparing the results for all algorithms,
our proposed EWLS outperforms its peers.
7 Conclusions
The hop-distance relationship information can effectively
improve the performance of the protocols for wireless
sensor networks in many aspects. However, most studies
focus on the UDG model which significantly deviates
from the real world. In the paper, we presented an analy-
tical modeling to formulate the hop-distance relationship
considering the quasi-UDG model. Senor nodes are ran-
domly distributed in a circular region according to a
Poisson point process. The probability of a particular hop

count given a known distance Ω
h
(d) was studied, and the
border effect and dependence problem was considered in
our analysis. Precisely, we derived the probability density
function of a random variable describing the distance
between two arbitrary nodes with a given hop count.
(a) Quasi-UDG Network (b) LS
(
c
)
PDM
(
d
)
EWLS
Figure 7 Localization error distributions on the quasi-UDG network topology.
Gao et al. EURASIP Journal on Wireless Communications and Networking 2011, 2011:99
/>Page 9 of 11
Simulation results confirmed that our analytical results
gave excellent accuracy. From the results, we further illu-
strated impact of the border effect.
Furthermore, we demonstrated the application of our
developed hop-distance relationship considering the
quasi-UDG model in WSN localizations. We designed a
LS-based localization algorithm using our developed
relationship and compared its performance with other
popular LS-based localization algorithms. We again con-
firmed that the explicit use of our developed relation-
ship in the computation of localization algorithms

improved the localization accuracy.
A Appendix
Suppose that a node x(x, y) is randomly deployed in a
circular region with the radiu s R
b
, the joint distribution
f
x
(x, y) can be obtained from
f
x
(x, y)=

1
πR
2
b
, x
2
+ y
2
≤ R
2
b
0, elsewhere
.
(25)
As the n odes x
1
(x

1
, y
1
)andx
2
(x
2
, y
2
) are selected
independently, the joint pdf of x
1
and x
2
is
f
x
1
,x
2
(x
1
, y
1
, x
2
, y
2
)=


1
(π R
2
b
)
2
, x
2
i
+ y
2
i
≤ R
2
b
, i =1,2
0, e1sewhere
.
(26)
We set x
d
= x
1
- x
2
and x
m
=(x
1
+ x

2
)/2. The joint
distribution of x
m
and x
d
can be obtained as
f
x
d
,x
m
(x
d
, y
d
, x
m
, y
m
)=

1
(π R
2
b
)
2
x
d

, x
m
∈ L
1
∩ L
2
0, elsewhere
(27)
where the constraints L
1
and L
2
are
L
1
:(x
m
+ x
d
/2)
2
+(y
m
+ y
d
/2)
2
< R
2
b

L
2
:(x
m
− x
d
/2)
2
+(y
m
− y
d
/2)
2
< R
2
b
.
(28)
We set the probability of the geographical distance
D
between x
1
and x
2
less than d to be
P(D ≤ d)
,andthe
constraint L
3

can be expressed by
L
3
: D
2
= x
2
d
+ y
2
d
≤ d
2
,
then we have
P(D ≤ d)

L1∩L
2
∩L
3
f
X
d
,X
m
(x
d
, y
d

, x
m
, y
m
)dx
m
dy
m
dx
d
dy
d
.
(29)
With L
1
∩ L
2
,thenx
m
falls into the intersectional
region of two circles with centers (x
d
/2, y
d
/2) and (-x
d
/
2, -y
d

/2). The intersectional area is
2R
2
b
arccos
⎛
⎜
⎝

x
2
d
+ y
2
d
2R
b
⎞
⎟
⎠
−

x
2
d
+ y
2
d
×





R
2
b
−

x
2
d
+ y
2
d
4

.
(30)
Figure 8 Effect on the average localization error ξ of anchor
fraction r
a
.
Figure 9 Effect on the average localization error of quasi-UDG
factor a.
Figure 10 Effect on the average localization error ξ of nodes’
communication range R.
Gao et al. EURASIP Journal on Wireless Communications and Networking 2011, 2011:99
/>Page 10 of 11
Since
f

x
d
,x
m
(x
d
, y
d
, x
m
, y
m
)
is constant, (29) can be
rewritten as
P(D ≤ d)=
1
πR
4

d
0

4R
2
arccos (
l
2R
) − l


4R
2
− l
2

ld
l
(31)
Therefore, we have
f
D
(d)=
d
πR
4

4R
2
arccos

d
2R

− d

4R
2
− d
2


(32)
where 0 <d <2R
b
.
Acknowledgements
The authors gratefully acknowledge the support of the Program of
Introducing Talents of Discipline to Universities ("111 Project”) under grant
No. B08002, and the support of the National Natural Science Foundation of
China (NSFC) under Grants No. 60802016, 60833002 and 60972010, the
support by “the Fundamental Research Funds for the Central Universities”
under grant No. 2009JBM007.
Author details
1
School of Electronics and Information Engineering, Beijing Jiaotong
University, Beijing 100044, PR China
2
TEDA College, Nankai University, Tianjin
300457, PR China
3
School of Computer Engineering, Nanyang Technological
University, 639798, Singapore
Competing interests
The authors declare that they have no competing interests.
Received: 31 December 2010 Accepted: 17 September 2011
Published: 17 September 2011
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doi:10.1186/1687-1499-2011-99
Cite this article as: Gao et al.: Hop-distance relationship analysis with
quasi-UDG model for node localization in wireless sensor networks.
EURASIP Journal on Wireless Communications and Networking 2011 2011:99.
Gao et al. EURASIP Journal on Wireless Communications and Networking 2011, 2011:99
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